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MTH/221 Week 1 - Connect Exercises Combinatorics

1.1 Apply basic enumeration techniques.

1.2 Apply basic permutation and combination techniques.

1.3 Apply introductory probability techniques.

Supporting Activity

What is the difference between combinations and permutations? What are some practical

applications of combinations? Permutations?

Supporting Activity

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Find the number of permutations of A,B,C,D,E,F taken three at a time (in other words find the

number of "3-letter words" using only the given six letters WITHOUT repetition).

Supporting Activity

Suppose you are assigning 6 indistinguishable print jobs to 4 indistinguishable printers. In how

many ways can the print jobs be distributed to the printers?

Supporting Activity

Suppose UOPX has 3 different math courses, 4 different business courses, and 2

different sociology courses. Tell me the number of ways a student can choose one

of EACH kind of course. Then tell me the number of ways a student can choose

JUST one of the course.

Supporting Activity

Consider the problem of how to arrange a group of n people so each person can

shake hands with every other person. How might you organize this process? How

many times will each person shake hands with someone else? How many

handshakes will occur? How must your method vary according to whether or not

n is even or odd?

WEEK 1 CONNECT EXERCISES AND DETAILED ANSWERS

NOTE: Even if your questions are not exactly the same, the

explanations in this tutorial should be enough to help you get the

correct answers!

1. A particular brand of shirt comes in 13 colors, has a male version and a female

version, and comes in 3 sizes for each sex. How many different types of this shirt

are made?

2. How many strings of five decimal digits

1. do not contain the same digit twice?

2.

end with an even digit?



3. have exactly four digits that are 9’s?

3. How many strings of six uppercase English letters are there

1.

if letters can be repeated?

2. if no letter can be repeated?

3. that start with X, if letters can be repeated?

4.

that start with X, if no letter can be repeated?

5. that start and end with X, if letters can be repeated?

6. that start with the letters NE (in that order). if letters can be repeated?

7.

that start and end with the letters NE (in that order), if letters can be

repeated?

8. that start or end with the letters NE (in that order), if letters can be

repeated?

4. In how many different orders can five runners finish a race if no ties are

allowed?

5. How many bit strings of length 9 have

exactly three O s?

more O s than 1 s?

at least six 1 s?

at least three 1 s?

6. A club has 16 members

a) How many ways are there to choose four members of the club to serve

on an executive committee?

b) How many ways are there to choose a president. vice president.

secretary. and treasurer of the club, where no person can hold more than

one office?

7. Five women and nine men are on the faculty in the mathematics department at

a school

a) How many ways are there to select a committe of five members of the

department if at least one woman must be on the committee?

b) How many ways are there to select a committee of five members of the

department if at least one woman and at least one man must be on the

committee?



8. Find the coefficient of in x16y4 in (x + y)20

9. What is the coefficient of x8 in (3 + x)12 ?

10. In how many different ways can seven elements be selected in order from a

set with four elements when repetition is allowed?

11. How many ways are there to assign three jobs to twenty employees if each

employee can be given more than one job?

12. How many solutions are there to the equation

x1+x2+x3+x4+x5 =21

where xi , i = 1, 2, 3, 4, 5, is a nonnegative integer such that

a) x1 ≥ 1?

b) x1 ≥ 3 for i= 1,2,3,4,5?c) 0 ≤ x1 ≤ 3, 1 ≤ x2 < 4, and x3 ≥ 15?

d) 0 ≤ x1 ≤ 3, 1 ≤ x2 < 4, and x3 ≥ 15?

13. How many ways are there to distribute thirteen indistinguishable balls into eight

distinguishable ins?

14. How many different strings can be made from the letters in MISSISSIPPI. using all

the letters?

15. What is the probability that a five-card poker hand contains the nine of

diamonds, the eight of clubs and the king of spades?

16. What is the probability that a fair die never comes up an even number when it

is rolled four times?

17. Find the probability of selecting none of the correct six integers in a lottery,

where the order in which these integers are selected does not matter, from the

positive integers not exceeding 5O.

18. What is the probability that Bo, Colleen, Jeff, and Rohini win the first, second,

third, and fourth prizes, respectively, in a drawing if 48 people enter a contest and

DETAILED ANSWERS PROVIDED



MTH/221 Week 2 - Connect Exercises

(Includes Bonus 2015 Connect Exercises and Answers!!)

Logic & Set Theory; Boolean Algebra; Relations & Functions

2.1 Use truth tables for propositional logic.

2.2 Simplify assertions and compound statements in first-order logic.

2.3 Apply basic set-theoretic concepts.

2.4 Use a Venn Diagram to visualize set relationships.

2.5 Differentiate between relations and functions.

2.6 Apply the basic concepts of Boolean algebra.

Supporting Activity

There is an old joke, commonly attributed to Groucho Marx, which goes

something like this: "I don't want to belong to any club that will accept people like

me as a member." Does this statement fall under the purview of Russell's paradox,

or is there an easy semantic way out? Look up the term fuzzy set theory in a

search engine of your choice or the University Library, and see if this theory can

offer any insights into this statement.

Supporting Activity

How do we distinguish relations from functions?

Supporting Activity











What sort of relation is friendship, using the human or sociological meaning of the word? Is it

necessarily reflexive, symmetric, antisymmetric, or transitive? Explain why it is or is not any of

these. What other types of interpersonal relationships share one or more of these properties?

Explain.

Supporting Activity

Write the dual of the following Boolean equation: a+a'b = a+b?

Supporting Activity

Reduce the following Boolean product to zero OR a fundamental product: xyx'z.

WEEK 2 CONNECT EXCERCISES (Includes Bonus 2015 Connect Exercises and Answers!!)

NOTE: Even if your questions are not exactly the same, the

explanations in this tutorial should be enough to help you get the

correct answers!

1. Which of these are propositions? What is the truth value of those that are propositions?

a) Answer this question.

b) What time is it?

c) Miami is the capital of Florida.

d) 6+x=9

e) The moon is made of green cheese.

f) 2n≥130

#2. Let p, q, and r be the propositions

p: You have the flu.

q: You miss the final examination.

r: You pass the course.

Express each of these propositions as an English sentence.





a) ¬q→¬p

o You miss the final examination if you do not have the flu.

o If you do not miss the final examination, then you do not have the flu.

o If you miss the final examination, then you have the flu.

o If you miss the final examination, then you do not have the flu.

o

If you do not miss the final examination, then you have the flu.

b) ¬q↔p

o You do not miss the final examination if and only if you do not have the flu.

o You do not miss the final examination if and only if you have the flu.

o You do not miss the final examination if you have the flu.

o You miss the final examination if you do not have the flu.

o You miss the final examination if and only if you do not have the flu.

c) p q r

o You have the flu, or miss the final exam, or pass the course.

o You have the flu, and miss the final exam, and pass the course.

o You have the flu, and miss the final exam, or pass the course.

o

You have the flu, or miss the final exam, and pass the course.

o You do not have the flu, or miss the final exam, or pass the course.

d) (p→¬r)

(q→¬r)

o If you do not pass the course, then you have the flu and you missed the final exam.

o It is the case that if you do not pass the course, then you have the flu or missed the

final exam.

o It is the case that if you have the flu and miss the final exam, then you do not pass

the course.

o It is either the case that if you have the flu then you do not pass the course or the

case that if you miss the final exam then you do not pass the course.

o

It is the case that if you have the flu then you do not pass the course and the casethat if you miss the final exam then you do not pass the course.

e) (p q) (¬q r)

o Either you have the flu or miss the final exam, or you do not miss the final exam and

pass the course.

o

Either you have the flu and miss the final exam, or you do not miss the final exam

and pass the course.



o You have both the flu and miss the final exam, and do not miss the final exam and

pass the course.

o You have either the flu or miss the final exam, or you do not miss the final exam or

pass the course.

o You have the flu or miss the final exam, and you do not miss the final exam or pass

the course.

#3. State the converse, contrapositive, and inverse of each of these conditional statements.

a) If it snows today, I will ski tomorrow.

b) I come to class whenever there is going to be a quiz.

c) A positive integer is a prime only if it has no divisors other than 1 and itself.

#4. Complete the truth table for each of these compound propositions.

a) p→(¬q∧r )

b) ¬ p→(q→r )

c) ( p→q)∨(¬ p→r )

d) ( p↔q)∧(¬q↔r )

e) (¬ p↔¬q)↔(q↔r )

#5. Are these system specifications consistent?

"Whenever the system software is being upgraded, users cannot access the file system. If users canaccess the file system, then they can save new files. If users cannot save new files, then the system

software is not being upgraded."

Let the following statements be represented symbolically as shown:

u: "The software system is being upgraded."

a: "Users can access the file system."

s: "Users can save new files."

Write each system specification symbolically.

"Whenever the system software is being upgraded, users cannot access the file system."

u∧¬a

u→a

u↔¬a

¬a→u



u→¬a

"If users can access the file system, then they can save new files."

a∨ s

a∧ s

s→a

a↔ s

a→ s

"If users cannot save new files, then the system software is not being upgraded."

¬u→¬ s

¬ s∧¬u

¬ s↔¬u

¬ s→¬u

¬ s→u

Is the system consistent?

No, this system is not consistent.

Yes, for example making v false, a false, and s true makes it consistent.

Yes, the conditional statements are always true.

#6. An explorer is captured by a group of cannibals. There are two types of cannibals – those who

always tell the truth and those who always lie. The cannibals will barbecue the explorer unless she

can determine whether a particular cannibal always lies or always tells the truth. She is allowed to

ask the cannibal exactly one question.

a) Explain why the question “Are you a liar?” does not work.

Both types of cannibals will answer with "no".

Both types of cannibals will answer with "yes".



The incorrect conclusion that the cannibal is one who always tells the truth will be made if

the answer is "no".

The incorrect conclusion that the cannibal is one who always tells the truth will be made if

the answer is "yes".

b) Which of the following questions does work in determining whether the cannibal she is speaking

to is a truth teller or a liar?

Select all the questions that work.

If I were to ask you if you always told the truth, would you say that you did?

If I say that you are a truth teller, would I be correct?

Do you always tell the truth?

Is the color of the sky blue?

If I say that you are a liar, would I be correct?

#7. Use De Morgan's laws to find the negation of the following statement.

James is young and strong.

James is young or he is not strong.

James is not young and he is not strong.

James is young or he is strong.


James is not young or he is not strong.

#8. Show that each of these conditional statements is a tautology by completing the truth tables.

a) (p q )→q

q ∧q ( p∧q)→q

T T



T F

F T

F F

b) p →(p q )

q ∨q →( p∨q)

T T

T F

F T

F F

c) ¬p →(p →q )

q ¬ p →q ¬ p→( p→q)

T T

T F

F T

F F

d) (p q )→(p →q )

q ∧q →q ( p∧q)→( p→q)

T T

T F

F T

F F

e) ¬(p →q )→p

q →q ( p→q) ¬( p→q)→ p

T T



T F

F T

F F

f) ¬(p →q )→¬q

q →q ¬( p→q) ¬q ¬( p→q)→¬q

T T

T F

F T

F F

We conclude that each of these conditional statements is a tautology because____________

#9. Use set builder notation to give a description of each of these sets.

a) {0,4,8,12,16}

{4n|n∈Z}

{4n|n≤16}

{n|n≤16}

{4n|n=1,2,3,4}

{4n|n=0,1,2,3,4}

b) {−2, −1, 0, 1, 2}

{ x|−2≤ x≤2}, where the domain is the set of integers.

{ x|−2≤ x≤2}



{ x|−2< x<2}, where the domain is the set of real integers.

{ x|−2≤ x≤2}, where the domain is the set of real numbers.

{ x|−2≤ x≤2}, where the domain is the set of natural numbers.

c) {a, b, c, r, s}

{ x| x is a letter of the word scrabble other than l or e}

{ x| x is a letter}

{ x| x is a letter of the word scrabble}

{ x| x is a letter of the word scrabble other than l }

{ x| x is a letter of the word scrabble other than e}

#10. Determine whether each of these pairs of sets are equal.

a) {1, 1, 3, 3, 5, 5, 5, 6},{5, 3, 1}

b) {{3}},{3,{3}}

c) ∅,{∅}

#11. Suppose that A={2,5,6}, B={2,6}, C ={5,6}, and D={5,6,7}. Determine which of these sets are

subsets of which other of these sets.

A⊆

A

B

C

D

B⊆

A



B

C

D

C ⊆

A

B

C

D

D⊆

A

B

C

D

#12. Use a Venn diagram to illustrate the relationship A⊆ B and B⊆C .



ALL DETAILED ANSWERS PROVIDED

MTH/221 Week 3 - Connect Exercises

(Includes Bonus 2015 Connect Exercises and Answers!!)

Algorithmic Concepts

3.1 Apply the basic concepts of algorithmic analysis.

3.2 Apply the introductory principles of mathematical induction.

3.3 Solve problems of iteration and recursion.

Supporting Activity

Describe a situation in your professional or personal life when recursion, or at least

the principle of recursion, played a role in accomplishing a task, such as a large

chore that could be decomposed into smaller chunks that were easier to handle

separately, but still had the semblance of the overall task. Did you track the

completion of this task in any way to ensure that no pieces were left undone, much

like an algorithm keeps placeholders to trace a way back from a recursive

trajectory? If so, how did you do it? If not, why did you not?

Supporting Activity

Given this recursive algorithm for computing a factorial...

procedure factorial(n: nonnegative integer)











if n = 0 then return 1

else return n *factorial(n − 1)

{output is n!} Show all the steps used to find 5!

Supporting Activity

Describe an induction process. How does induction process differ from a process of simplerepetition?

Supporting Activity

List all the steps used to search for 9 in the sequence 1,3, 4, 5, 6, 8, 9, 11 using a binary search.

WEEK 3 CONNECT EXERCISES

NOTE: Even if your questions are not exactly the same, the explanations

in this tutorial should be enough to help you get the correct answers!

1. List all the steps used by Algorithm 1 to find the maximum of the list

3,9,14,7,11,4,18,3,11,2

2. Determine which characteristics of an algorithm described in the text (after Algorithm 1) the

following procedures have and which they lack. Select characteristics that the procedures have and

leave characteristics unselected that the procedures lack.

a. procedure double (n: positive integer)

while n > 0

n = 3n

b. procedure divide(n: positive integer)

while n ≥ 0

m = 1 / n n = n − 1

c. procedure sum(n: positive integer)

sum = 0

while I < 3

sum = sum + i





d. procedure choose(a,b: integers)

= either a or b

3. Which of the following algorithms can be used to find the sum of all the integers in a list?

4. Which statement best describes an algorithm that takes as input a list of n integers and

produces as output the smallest difference obtained by subtracting an integer in the list from the one

following it?

Set the answer to be ∞. For i going from 1 through n−1, compute the

value of the (n+1)st element in the list minus the nth element in the list.

If this is smaller than the answer, reset the answer to be this value.

Set the answer to be 0. For i going from 1 through n, compute the valueof the (i+1)st element in the list minus the ith element in the list. If this

is smaller than the answer, reset the answer to be this value.

Set the answer to be −∞. For i going from 1 through n−1, compute the

value of the (i+1)st element in the list minus the ith element in the list.


Set the answer to be0. For i going from 1 through n−1, compute the



Set the answer to be ∞. For i going from 1 through n−1, compute the



5. Consider an algorithm that uses only assignment statements that replaces the hextuple (u, v,

w, x, y, z ) with (v, w, x, y, z, u). What is the minimum number of assignment statements needed?

6. Which of the following is an algorithm that locates the last occurrence of the smallest element

in a finite list of integers? The integers in the list are not necessarily distinct.

7. To establish a big-O relationship, find witnesses C and k such that | f ( x)|≤C | g ( x)| whenever x>k .

Determine whether each of these functions is O( x^ 2).

8. To establish a big-O relationship, find witnesses C and k such that | f ( x)|≤C | g ( x)| whenever x>k .



9. k be a positive integer. Which of the following mathematical equations correctly

shows

that 1k +2k +⋅⋅⋅+nk is O(nk +1)?

1k +2 k +⋅⋅⋅+n k ≤1⋅n k +2⋅nk +⋅⋅⋅+n⋅nk ≤(1+2+⋅⋅⋅+n)⋅nk +1≤nk +1

1k +2 k +⋅⋅⋅+n k ≤n k +1+nk +1+⋅⋅⋅+nk +1=nk +1

1k +2 k +⋅⋅⋅+nk ≤1k ⋅n+2k ⋅n+⋅⋅⋅+nk ⋅n=n⋅(1k +2k +⋅⋅⋅+(n−1)k )+n⋅nk ≤nk +1

1 k +2 k +⋅⋅⋅+n k ≤(1+2+⋅⋅⋅+n)k ≤nk +1

1 k +2 k +⋅⋅⋅+n k ≤n k +n k +⋅⋅⋅+n k =n⋅n k =n k +1

10. Give a big-O estimate for each of the following functions. For the function g in the

estimate

( x)is O(g( x )), use a simple function g of smallest order.

a) (n

5

+ n

4

log n ) (log n + 1) + (14 log n + 18)(n

5

+ 11)

g(n)=

b) (3n + n 3)(n 2 + 2n )

g>(n )=

c) (n n + n 7n + 9n )(n ! + 9n )

g(n)=

11. Give a big-O estimate for the number of comparisons used by the algorithm that

determines the number of 1s in a bit string of length n by examining each bit of the

string to determine whether it is a 1 bit.



The number of comparisons is O ( ).

12. There is a more efficient algorithm (in terms of the number of multiplications and additions

used) for evaluating polynomials than the conventional algorithm. It is called Horner's method.

This pseudocode shows how to use this method to find the value of a n x n + a n −1x n −1 +⋅⋅⋅+ a 1x + a 0

at x = c .

procedure Horner (c ,a 0,a 1,a 2,...,an : real numbers)

y a n

for I 1 to n

y = y * c+a n−i

return y{y = a n c n + a n −1 c n −1 +⋅⋅⋅

+ a 1c +a 0}

a) Evaluate 2x 2+4x +5 at x =5 by working through each step of the algorithm showing the

values assigned at each assignment step.

y =

i =

y =

i =

y =

b) Exactly how many multiplications and additions are used to evaluate a

polynomial of degree

n at x = c ? (Do not count additions used to increment the loop variable.)

There are multiplications.

There are additions.



13. What is the largest n for which one can solve within one second a problem using an

algorithm that requires f (n ) bit operations, where each bit operation is carried out in 10−6

seconds, with these functions f (n )?

For questions (a) to (c), enter the exact answers. Enter brackets around exponents. Round

your answers down to the nearest integer for all other parts. Note that log n = log2 n .

a) Log n<

n =

b) n

c) n * log n

nn=

d) n2

n =

e) 2 n

n =

f) n!

n =

14. How much time does an algor ithm take to solve a problem of size n if thi s algor ithm uses 2n 2 +

2 n operations, each requir ing 10 −7 seconds, with the fol lowing values of n?

Round your answers to three sign if icant digits. Enter very large or very small numbers using

scienti fi c notation.

How much time does an algor ithm take to solve a problem of size n i f thi s algori thm uses 2n2 + 2n

operations, each requiring 10−7 seconds, with the following values of n?

Round your answers to three signif icant digits. Enter very large or very small numbers using

scienti fi c notation.



a) n=10

_______seconds

b) n=20

_______seconds

c) n=50

_______seconds

d) n=100

________seconds

15. Determine the least number of comparisons, or best-case per formance,

a) required to find the maximum of a sequence of n in tegers, using Al gori thm 1 of Section 3.1.

O( )

b) used to locate an element i n a l ist of n terms with a li near search.

O( )

c) used to locate an element in a li st of n terms using a binary search.

O( )

16. Let P(n) be the statement that13+23+···+n3=(n(n+1)/2)2 for the positi ve in teger n .a)

What is the statement P(1)?P(1) i s the statement

13=[2 * 1+1/2]2

13=[1 * (0+1)2/2]2

13=12

13=[1 * (1+2)/2]2

13=[1 *( 1+1)/2]2

b) Show that P(1) is true, completing the basis step of the proof.

The left-hand side of the basis step is ____________ .



The right-hand side of the basis step is___________ .

c) What is the inductive hypothesis?

The inductive hypothesis is the statement that 13+23+···+k 3= .

d) What do you need to prove in the inductive step?

You need to prove that [13

+23

+···+k 3

]+(k+1)3

for all k ≥ .

e) Complete the inductive step, identi fying where you use the inducti ve hypothesis.

The inductive hypothesis replaces the quanti ty k3 f rom the left-hand side of par t (d),

which shows that [13+23+···+k 3 ]+(k+1)3=(k+1)(k+2)/2)2

The inductive hypothesis replaces the quanti ty k+1 f rom the left -hand side of part

(d), which shows that [13+23+···+k 3 ]+(k+1)3=(k+1)(k+2)/2)2

The inductive hypothesis replaces the quanti ty 13+23+ +k3 fr om the left-hand

side of part (d), which shows that [ [13+23+···+k 3 ]+(k+1)3=(k+1)(k+2)/2)2

The inductive hypothesis replaces the quanti ty k from the left-hand side of part (d),

which shows that [13+23+···+k 3 ]+(k+1)3=(k+1)(k+2)/2)2

The inductive hypothesis replaces the quanti ty (k+1)3 f rom the left-h and side of

part (d), whi ch shows that [13+23+···+k 3 ]+(k+1)3=(k+1)(k+2)/2)2

f ) Explain why these steps show that thi s formula is true whenever n is a posit ive integer .

We have completed both the basis step and the inductive step, so by the principle of mathemati cal

induction, the statement i s true for every integer n.

We have completed both the basis step and the inductive step, so by the principle of mathemati cal

induction, the statement is true for every positi ve in teger n.



We have completed the inductive step, so by the principle of mathemati cal induction , the statement

is true for every positi ve in teger n.

We have completed the inductive step, so by the principle of mathemati cal induction , the statement

is true for every integer n.

We have completed both the basis step and the inductive step, so by the principle

of mathematical induction, the statement is true for every real number n .

PLUS QUESTIONS 17 THROUGH 28 WITH DETAILED ANSWERS!!



MTH/221 Week 4 -Connect Exercises

Graph Theory and Trees

4.1 Apply properties of general graphs.

4.2 Apply properties of trees.

Supporting Activity: Hamiltonian and Euler Graphs

Note a Hamiltonian circuit visits each vertex only once but may repeat edges. A

Eulerian graph traverses every edge once, but may repeat vertice.

Looking at the figure below tell me if they are Hamiltonian and/or Eulerian.

*-------*--------*

| / | \ |

| / | \ |

|/ | \ |

*-------*--------*











Supporting Activity: Path Analysis Class

Here is a Question for you:

For the diagram below find all the "simple paths" from A to F.

A------------B------------C

| | / |

| | / |

| | / |

D------------E------------F

Supporting Activity: Random Graphs

Random graphs are a fascinating subject of applied and theoretical research. These can be generated

with a fixed vertex set V and edges added to the edge set E based on some probability model, such as

a coin flip. Speculate on how many connected components a random graph might have if the

likelihood of an edge (v1,v2) being in the set E is 50%. Do you think the number of components

would depend on the size of the vertex set V? Explain why or why not.

Supporting Activity: Trees and Language Processing

Trees occur in various venues in computer science: decision trees in algorithms, search trees, and so

on. In linguistics, one encounters trees as well, typically as parse trees, which are essentially sentence

diagrams, such as those you might have had to do in primary school, breaking a natural-language

sentence into its components--clauses, subclauses, nouns, verbs, adverbs, adjectives, prepositions, and

so on. What might be the significance of the depth and breadth of a parse tree relative to the sentence

it represents? If you need to, look up parse tree and natural language processing on the Internet to see

some examples.

WEEK 4 CONNECT EXERCISES QUESTIONS AND ANSWERS TO #1 to #14



#1. Determine whether the graph shown has directed or undirected edges, whether it has multiple

edges, and whether it has one or more loops. Use your answers to determine the type of graph in

Table 1 this graph is.





#2. The intersection graph of a collection of sets A1, A2, . . . , An is the graph that has a vertex for

each of these sets and has an edge connecting the vertices representing two sets if these sets have a

nonempty intersection. Select the intersection graph of these collections of sets.

#3. Find the number of vertices, the number of edges, and the degree of each vertex in the given

undirected graph. Identify all isolated and pendant vertices.

v=

e=

Enter the degree of each vertex as a list separated by commas, starting from vertex a and proceeding

in alphabetical order.

Enter the isolated vertices as a list separated by commas, starting from vertex a and proceeding in

alphabetical order. Enter NA if there is no isolated vertex.

Enter the pendant vertices as a list separated by commas, starting from vertex a and proceeding in

alphabetical order. Enter NA if there is no pendant vertex.

#4. Determine the number of vertices and edges and find the in-degree and out-degree of each

vertex for the given directed multigraph.



v=

e=

deg−(a)= deg+(a)=

deg−(b)=

deg+(b)=

deg−(c)=

deg+(c)=

deg−(d)=

deg+(d)=

#5. Use an adjacency list to represent the given graph.

Enter the vertices in alphabetical order, separated by commas.

#6. Use an adjacency matrix to represent the given graph. Assume the vertices are listed in

alphabetical order.

#7. Which of the following graphs has the given adjacency matrix?



INCLUDES REMAINING QUESTIONS AND ANSWERS FROM

#8 to #14

MTH/221 - Week 5

Final Exam & Discussion Questions

Applications of Discrete Mathematics

NOTE: The Week 5 Tutorials include ALL Connect Exercise Answers fromWeek 1, Week 2, Week 3, Week 4, and Week 5. Even if your questions

are not exactly the same, the explanations in this tutorial should be

enough to help you get the correct answers!

Option 2: Coding Theory Case StudyExplain the theory in your own words based on the case study and suggested

readings. Include the following in your explanation:

Error Detecting Codes

Error Correcting Codes

Hamming Distance

Perfect Codes

Generator Matrices

Parity Check Matrices

Hamming Codes










Give an example of how this could be applied in other real-world

applications. Format your paper according to APA guidelines. All work must be

properly cited and referenced.

DISCRETE MATH - FINAL EXAM - CONNECT

EXERCISES

ALL DETAILED ANSWERS PROVIDED

NOTE: The Week 5 Tutorials include ALL Connect Exercise Answers from

Week 1, Week 2, Week 3, Week 4, and Week 5. Even if your questions

are not exactly the same, the explanations in this tutorial should be

enough to help you get the correct answers!

1. A particular brand of shirt comes in 8 colors, has a male version and a female

version, and comes in 5 sizes for each sex. How many different types of this shirt are

made?

2. A club has 22 members.

a) How many ways are there to choose four members of the club to serve on an

executive committee?

b) How many ways are there to choose a president, vice president, secretary, and

treasurer of the club, where no person can hold more than one office?

3. In how many different ways can ten elements be selected in order from a set

with four elements when repetition is allowed?

4. What is the probability that a fair die never comes up an odd number when it is

rolled eight times?

5. Let p, q, and r be the propositions

p:You have the flu.q:You miss the final examination.

r : You pass the course.

Express each of these propositions as an English sentence.

a) ¬q →r



You miss the final examination if you pass the course.

If you do not miss the final examination, then you pass the course.

If you miss the final examination, then you pass the course.

If you do not miss the final examination, then you do not pass the course.

If you miss the final examination, then you do not pass the course.

b) ¬q ↔p

You miss the final examination if and only if you do not have the flu.

You do not miss the final examination if you have the flu.

You do not miss the final examination if and only if you do not have the flu.

You do not miss the final examination if and only if you have the flu.

You miss the final examination if you do not have the flu.

c) p q r

You have the flu, or miss the final exam, and pass the course.

You do not have the flu, or miss the final exam, or pass the course.

You have the flu, and miss the final exam, and pass the course.

You have the flu, and miss the final exam, or pass the course.

You have the flu, or miss the final exam, or pass the course.

d) (p →¬r ) (q →¬r )

It is the case that if you have the flu and miss the final exam, then you do not pass the course.

It is the case that if you have the flu then you do not pass the course and the case that if you miss the

final exam then you do not pass the course.

It is the case that if you do not pass the course, then you have the flu or missed the final exam.



It is either the case that if you have the flu then you do not pass the course or the case that if you miss

the final exam then you do not pass the course.

If you do not pass the course, then you have the flu and you missed the final exam.

e) (p

q )

(¬q

r )

You have either the flu or miss the final exam, or you do not miss the final exam or pass the course.

Either you have the flu and miss the final exam, or you do not miss the final exam and pass the course.

You have both the flu and miss the final exam, and do not miss the final exam and pass the course.

Either you have the flu or miss the final exam, or you do not miss the final exam and pass the course.

You have the flu or miss the final exam, and you do not miss the final exam or pass the course.

6. Complete the truth table for each of these compound propositions.

a) p →(¬q r )

p q r ¬q ¬q∧r p →(¬q

r )

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

b) ¬p →(r →q )

p q r ¬ p r →q ¬p →(r →q )

T T T

T T F

T F T

T F F

F T T



F T F

F F T

F F F

c) (p →q ) (¬p →r )

p q r ¬ p →q ¬ p→r (p →q ) (¬p →r )

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

d) (p ↔q )

(¬q ↔r )

p q r ¬q ↔q ¬q↔r (p ↔q )

(¬q ↔r )

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

e) (¬p ↔¬q )↔(q ↔r )

p q r ¬ p ¬q ¬ p↔¬q q↔r (¬p ↔¬q )↔(q ↔r )

T T T

T T F

T F T

T F F

F T T



F T F

F F T

F F F

7. Use De Morgan's laws to find the negation of the following statement.

James is young and strong.

James is young or he is strong.



James is young or he is not strong.

James is not young or he is not strong.

8. Show that each of these conditional statements is a tautology by

completing the truth tables.

a) (p q )→p

p q p q (p q )→p

T T

T F

F T

F F

b) q →(p q )

p q p q q →(p q )

T T

T F

F T

F F

c) ¬p →(p →q )



p q ¬p p →q ¬p →(p →q )

T T

T F

F T

F F

d) (p q )→(p →q )

p q p q p →q (p q )→(p →q )

T T

T F

F T

F F

e) ¬(p →q )→p

p q p →q ¬(p →q ) ¬(p →q )→p

T T

T F

F T

F F

f) ¬(p →q )→¬q

p q p → ¬(p →q ) ¬q ¬(p →q )→¬q

T T

T F

F T

F F

We conclude that each of these conditional statements is a tautology because the entries in the last

column contain _________.



9. Suppose that A={6,7,8}, B ={3,6,7}, C ={3,7}, and D ={6,7}. Determine which

of these sets are subsets of which other of these sets.

⊆

A

B

C

D

B⊆

A

B

C

D

C ⊆

A

B

C

D

D⊆

A

B

C

D

10. Use a Venn diagram to illustrate the relationship A⊆

B and B ⊆

C .



11. Suppose that A is the set of sophomores at your school and B is the set

of students in discrete mathematics at your school. Express each of these

sets in terms of A and B .

a) the set of sophomores at your school who are not taking discrete mathematics

∩ B —

∪ B

∪ B —

—∩ B

∩ B

b) the set of students at your school who either are sophomores or are taking discrete mathematics

— ∪ B —

∪ B —



∪ B

—∩ B —

∩ B

12. Let A={a ,b ,c ,d ,e } and B ={a ,b ,c ,d ,e ,f ,g ,h }. Find

a) A∩ B

{}

{ f , g ,h}

{a, b, c, d, e}

{a, b, c, d, e, f, g, h}

{a,b,c,d }

b) A− B

{}

{a,b,c,d ,e, f , g ,h}

{a,b,c,d ,e}

{e, f , g ,h}

{ f, g, h}

13. Select the correct Venn diagram for each of these combinations of the

sets A, B , and C .

a) B∪( A∩C )



b) A∩ B —∩C



c) ( B− A)∪(C − B)∪( A−C )

INCLUDES REMAINING QUESTIONS ANDANSWERS FROM #14 to #40

Supporting Activity



After performing some research or based on your reading in the course, share with

the class the most practical use of discrete mathematics (in your opinion). Please

cite your source.

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